Asymptotic behavior of equilibrium pair correlations in a dense electron gas

The nodal expansion of the potential of average force is worked out systematically by iterating all the nonconvolution compact graphs from the order at which they appear first to infinity. For this purpose, we perform an exhaustive study of the asymptotic behavior of the bridge (without articulation points) graphs, and show that they decrease as $\frac{\beta e^{-\alpha r}}{r}$, with $\alpha >1\mathrm{}$ and $r$ in units of the Debye length ${\lambda }_D$. We demonstrate that the asymptotic $w_2\mathrm{}\left(r\right)\mathrm{}$ expansion is obtained from the resummation of the longest convolution chains [$l\ge 2(n-1\mathrm{})\mathrm{}$] with $n-1\mathrm{}$ two-bubbles and $0\le c\le n$ Debye lines, which allow for a systematic improvement of the usual hypernetted-chain (HNC) approximation with the replacement of one, two, or more two-bubbles by bridge graphs. Substantial simplification of the final expression is achieved with the aid of the $n$-bubble sum which decreases asymptotically faster than the Debye line. The onset of short-range order is shown to arise at the critical value ${\Lambda }_c=4.247\mathrm{}$ of the plasma parameter $\Lambda =\frac{e^2}{k_B}T{\lambda }_D$ in excellent agreement with the Del Rio-DeWitt calculations.